In general, given your background, the most important thing you should do is proper networking. I wrote below my answers to your questions, and after that some tips on how to do that.
If I want to start from scratch, what consulting firms should I aim for?
I would search on LinkedIn where alumni from your universities are working now in consulting – this will give you an idea of the reasonable targets.
Is it possible to work in a lower tier consulting firm and try to apply for MBB after few years?
Definitely possible, it happens pretty often. Turnover in MBB is pretty high (average in many countries is 3 years), thus they continuously look for new hire, and former consultants are always a good subset where to look for (the brand of the lower tier consulting firm is obviously important for the screening).
Or should I start my career in other industries instead, and come back after few years?
If you want to do consulting, I would start now. Alternatives may be strong corporate brand in strategy role, but only if the corporate brand would be a lot stronger than the lower tier consulting firm one.
As for what you should do for networking, I would recommend the following:
1. Identify who are the people who could more easily help you. Alumni of your current university on LinkedIn should be the first targets
2. Write them a customized cold email
3. Have a call with the consultant, and indirectly ask for a referral
Of course, each step should be structured in a proper way to maximize the chances of moving to the next step. I have used this method when I applied for consulting, and I believe this is the main reason why I got invitations for all the applications I did (McK, Bain, BCG, Oliver Wyman, Roland Berger, Booz&Co - now Strategy&): I never applied through the website, as I always had a person referring me.
Finally, so far that you find alumni of your current university in MBB, it could be possible for you to be invited there as well with appropriate networking, thus I would try that for them as well.